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There is a symmetry between a function and its inverse. Specifically, if f is an invertible function with domain X and codomain Y, then its inverse f −1 has domain Y and image X, and the inverse of f −1 is the original function f. In symbols, for functions f:X → Y and f −1:Y → X, [13]
If an invertible function is C k with >, then so too is its inverse. This follows by induction using the fact that the map F ( A ) = A − 1 {\displaystyle F(A)=A^{-1}} on operators is C k for any k {\displaystyle k} (in the finite-dimensional case this is an elementary fact because the inverse of a matrix is given as the adjugate matrix ...
In calculus, the inverse function rule is a formula that expresses the derivative of the inverse of a bijective and differentiable function f in terms of the derivative of f. More precisely, if the inverse of f {\displaystyle f} is denoted as f − 1 {\displaystyle f^{-1}} , where f − 1 ( y ) = x {\displaystyle f^{-1}(y)=x} if and only if f ...
Functions that have inverse functions are said to be invertible. A function is invertible if and only if it is a bijection. Stated in concise mathematical notation, a function f: X → Y is bijective if and only if it satisfies the condition for every y in Y there is a unique x in X with y = f(x).
By convention, f 0 is defined as the identity map on f 's domain, id X. If Y = X and f: X → X admits an inverse function f −1, negative functional powers f −n are defined for n > 0 as the negated power of the inverse function: f −n = (f −1) n. [12] [10] [11]
In linear algebra, an invertible matrix is a square matrix which has an inverse. In other words, if some other matrix is multiplied by the invertible matrix, the result can be multiplied by an inverse to undo the operation. An invertible matrix multiplied by its inverse yields the identity matrix. Invertible matrices are the same size as their ...
More generally, the general linear group of degree n over any field F (such as the complex numbers), or a ring R (such as the ring of integers), is the set of n×n invertible matrices with entries from F (or R), again with matrix multiplication as the group operation. [1] Typical notation is GL n (F) or GL(n, F), or simply GL(n) if the field is ...
In other words, g is a right inverse of f if the composition f o g of g and f in that order is the identity function on the domain Y of g. The function g need not be a complete inverse of f because the composition in the other order, g o f, may not be the identity function on the domain X of f.